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The classical paper by Lighthill (Commun. Pure Appl. Maths, vol. 109, 1952, p. 118) on the propulsion of ciliated microorganisms has become the reference against which many modern studies on swimming in low Reynolds number are compared. However, Lighthill's study was limited to propulsion in a uniform flow, whereas several biologically relevant microorganisms experience non-uniform flows. Here we propose a benchmark for ciliary propulsion in paraboloidal flows. We first consider the axisymmetric problem, with the microorganisms on the centreline of the background flow, and derive exact analytical solutions for the flow field. Our results reveal flow features, swimming characteristics and performance metrics markedly different from those generated in a uniform flow. In particular, the background paraboloidal flow introduces a Stokes quadrupole singularity at the leading-order flow field, generating vortices. Moreover, we determine the necessary conditions on the strength of the background flow for optimal power dissipation and swimming efficiency. We then consider the more general case of a microorganism off the centreline of the background flow. In this case, the squirmer experiences a paraboloidal, linear shear and uniform flows due to its position relative to the flow's centreline. Our findings show that while the linear shear flow does not affect the translational and rotational velocities of the squirmer, it does influence the velocity field and, therefore, the power dissipation. 

Biological microorganisms and artificial micro-swimmers often locomote in heterogeneous viscous environments consisting of networks of obstacles embedded into viscous fluid media. In this work, we use the squirmer model and present a numerical investigation of the effects of shape on swimming in a heterogeneous medium. Specifically, we analyse the microorganism's propulsion speed as well as its energetic cost and swimming efficiency. The analysis allows us to probe the general characteristics of swimming in a heterogeneous viscous environment in comparison with the case of a purely viscous fluid. We found that a spheroidal microorganism always propels faster, expends less energy and is more efficient than a spherical microorganism in either a homogeneous fluid or a heterogeneous medium. Moreover, we determined that above a critical eccentricity, a spheroidal microorganism in a heterogeneous medium can swim faster than a spherical microorganism in a homogeneous fluid. Based on an analysis of the forces acting on the squirmer, we offer an explanation for the decrease in the squirmer's speed observed in heterogeneous media compared with homogeneous fluids. 

The development of novel drug delivery systems, which are revolutionizing modern medicine, is benefiting from studies on microorganisms’ swimming. In this paper we consider a model microorganism (a squirmer) enclosed in a viscous droplet to investigate the effects of medium heterogeneity or geometry on the propulsion speed of the caged squirmer. We first consider the squirmer and droplet to be spherical (no shape effects) and derive exact solutions for the equations governing the problem. For a squirmer with purely tangential surface velocity, the squirmer is always able to move inside the droplet (even when the latter ceases to move as a result of large fluid resistance of the heterogeneous medium). Adding radial modes to the surface velocity, we establish a new condition for the existence of a co-swimming speed (where squirmer and droplet move at the same speed). Next, to probe the effects of geometry on propulsion, we consider the squirmer and droplet to be in Newtonian fluids. For a squirmer with purely tangential surface velocity, numerical simulations reveal a strong dependence of the squirmer's speed on shapes, the size of the droplet and the viscosity contrast. We found that the squirmer speed is largest when the droplet size and squirmer's eccentricity are small, and the viscosity contrast is large. For co-swimming, our results reveal a complex, non-trivial interplay between the various factors that combine to yield the squirmer's propulsion speed. Taken together, our study provides several considerations for the efficient design of future drug delivery systems. 

A squirmer enclosed in a droplet represents a minimal model for some drug delivery systems. In the case of a spherical squirmer swimming with a spherical cage in a Newtonian fluid [Reigh et al., “Swimming with a cage: Low-Reynolds-number locomotion inside a droplet,” Soft Matter 13, 3161 (2017)], it was found that the squirmer and droplet always propelled in the same direction albeit at different speeds. We expand the model to include particles' shape and medium's heterogeneity, two biologically relevant features. Our results reveal a novel behavior: a configuration that consists of a spherical squirmer and a spheroidal droplet in highly heterogeneous media yields a backward motion of the droplet. 

Contaminants and other agents are often present at the interface between two fluids, giving rise to rheological properties such as surface shear and dilatational viscosities. The dynamics of viscous drops with interfacial viscosities has attracted greater interest in recent years, due to the influence of surface rheology on deformation and the surrounding flows. We investigate the effects of shear and dilatational viscosities on the electro-deformation of a viscous drop using the Taylor–Melcher leaky dielectric model. We use a large deformation analysis to derive an ordinary differential equation for the drop shape. Our model elucidates the contributions of each force to the overall deformation of the drop and reveals a rich range of dynamic behaviors that show the effects of surface viscosities and their dependence on rheological and electrical properties of the system. We also examine the physical mechanisms underlying the observed behaviors by analyzing the surface dilatation and surface deformation.